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In Nouredine Zettili's QM book, Hilbert space $H$ is said to be separable when: There exists a Cauchy sequence $\psi_n \in H$ ($n=1,2,\ldots)$ such that for every $\psi$ of $H$ and $\varepsilon > 0,$ there exists at least one $\psi_n$ of the sequence for which $||\psi - \psi_n || < \varepsilon.$

But in Kreyszig's Functional Analysis book, separable is defined to be: A metric space $X$ is said to be separable if it has a countable subset which is dense in $X$.

I tried to look for the separability of of Hilbert space in Kreyszig's but couldn't find anything similar to Zettili's. My question then is, that are the above mentioned two definitions of separable space equivalent?

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"There exists a Cauchy sequence ..." seems to be a typo (since Cauchy sequences are bounded, the condition is obviously nonsensical). Delete "Cauchy" and you get a dense sequence $\psi_n$, hence a countable dense set. For the other direction enumerate the countable dense set to get a sequence satisfying Zettili's condition. –  Martin Apr 24 '13 at 10:52
    
I am unable to see why this condition is nonsensical. Pls explain a little –  IPK Apr 25 '13 at 8:18
    
Since $\psi_n$ is supposed to be a Cauchy sequence, it is contained in some ball of radius $R$ around zero. Assuming $R \gt \varepsilon$, no vector $\psi$ of norm $\geq 2R$ can be near any $\psi_n$. –  Martin Apr 25 '13 at 10:51

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